Correction of Numerov's eigenvalue estimates

Andrew, Alan L.; Paine, John

doi:10.1007/bf01389712

Cited by 68 publications

(65 citation statements)

References 7 publications

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“…in his PhD Thesis (1979). Then it is followed up in various papers [13,14,16,19,20,26]. The correction terms can be added to the numerical eigenvalues in order to obtain better eigenvalues of the problem.…”

Section: Correction Termsmentioning

confidence: 99%

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Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Rattana¹,

Böckmann²

2013

Journal of Computational and Applied Mathematics

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Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Universitätsverlag AbstractThis paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.

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Section: Correction Termsmentioning

confidence: 99%

“…However, the two most common matrix methods are finite different method (FDM) and Numerov's method [12]. Correction terms of the methods for second order SLP have been described in [12,13,14,15,16,17,18,19,20] to obtain better numerical eigenvalues of the problem.…”

Section: Introductionmentioning

confidence: 99%

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Rattana¹,

Böckmann²

2013

Journal of Computational and Applied Mathematics

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“…been many developments in the basic approach of reduction to a matrix eigenproblem using finite differences and finite elements. Excellent surveys of such matrix methods are given in [2,3,4]. Matrix methods can only be used to approximate the first few eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Ledoux

Daele

Berghe

2009

Computer Physics Communications

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Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss some techniques that yield uniform approximation over the whole eigenvalue spectrum and can take large steps even for high eigenvalues. In particular, we will focus on methods based on coefficient approximation which replace the coefficient functions of the SturmLiouville problem by simpler approximations and then solve the approximating problem. The use of (modified) Magnus or Neumann integrators allows to extend the coefficient approximation idea to higher order methods.

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“…In many important cases [4,8,9], the correction is known in closed form. Originally proposed [22] for centered finite difference solution of the direct problem (1-2) with c 1 = c 2 = 0 , asymptotic correction was soon extended to higher order methods [11,25] and to more general differential operators [4,6,25].…”

Section: U3mentioning

confidence: 99%

“…Numerov's method [5,7,11] has the advantage that the matrices whose eigenvalues are required are tridiagonal, but the disadvantage that the mesh length is fixed by the number of available data points [5,7,8]. Gao et al [14] showed that this limitation could be overcome by using interpolation to refine the mesh.…”

Section: U3mentioning

confidence: 99%

Solving inverse Sturm-Liouville problems: theory and practice

Andrew

2017

ANZIAMJ

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Theoretical results on the solution of inverse Sturm-Liouville problems generally consider only idealized problems requiring much more data than is available in real applications. Typical theorems describe problems where infinitely many eigenvalues are known exactly, but in most applications we know only approximations of a finite, and usually small, number of eigenvalues. This paper considers how idealized theoretical results may assist practical numerical computation. It also reviews recent progress on a class of numerical methods for inverse Sturm-Liouville problems, it discusses some open questions, and it announces a new convergence result.

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Correction of Numerov's eigenvalue estimates

Cited by 68 publications

References 7 publications

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

Solving inverse Sturm-Liouville problems: theory and practice

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